Computers on the Qualifying Quiz

There are some problems in mathematics that no one has yet been
able to solve without a computer. Our quiz problems are not like
that! With each of them, once you find the right way of thinking
about it, a computer becomes unnecessary.

Of course, for those who like programming, a computer can be a good
way to come up with ideas or to test hypotheses. However, the
ultimate goal is to figure out the hidden pattern in each problem– to
understand what is really going on.

Eric and Matthew are visiting the town of Descartes,
Quibec, and have managed to lose each other. Descartes has five
equally spaced streets running east-west, cut by five equally spaced
streets running north-south; each of the 16 blocks is a perfect
square. Eric is at the southwest corner of town, and decides to look
for Matthew by walking towards the northeast corner along a path of
shortest length. At each intersection where he has a choice between
north and east he chooses which way to go randomly, with both
directions equally likely. Meanwhile, Matthew is at the northeast
corner, deciding to look for Eric by walking to the southwest corner
in exactly the same manner. Both start walking at the same time, and
walk at the same speed. What's the probability that they meet?

If you were to write a program to generate all possible paths that
Eric and Matthew could take and check in each case whether they meet,
you would be missing the point of the problem completely. The point
is to derive the probability from the general properties of such a
situation.

This is not just an annoying restriction we have dreamed up to make
your life more difficult. In real mathematics, it is often possible
to check certain special cases (such as a five-by-five grid of
streets) with the help of a computer, but the solution of a general
problem (such as an n-by-n grid of streets, for
arbitrary n) still practically always requires human insight –
and in almost all cases, it is the solution to the general problem,
rather than the specific one, that is more mathematically
interesting.

So feel free to use the computer in the initial stages of your
exploration. However, if your final solution still relies on a
computer, you are almost certainly missing some key
insight.